What characteristics are implied when you say that the machine operates at the beam-beam limit?

The beam-beam parameter (often also called beam-beam tune shift) characterizes the strength of the beam-beam force in a given plane on the particles of the opposing beam. The figues below show the deflection by the beam-beam force for the case of LEP (flat beams). Note the difference between x and y due to the fact that the horizontal beam size is much larger than the vertical size.

What is the ramp time?

It is the time to ramp the energy of the machine. For FCC-ee it is not relevant since FCC-ee operates at constant energy and the beam are topped-up continously from a booster ring. The booster ring has to deliver on average around 1012 e+ and e- per second. The booster is cycling its energy between its injection energy (12-40 GeV - to be defined) and the operating energy of FCC-ee. The cycle time is roughly 5 seconds - details to be worked out.

In what way does synchrotron radiation affect the correct performance of the machine?

This question can not really be answered with e+e- machines such SR is intrisic to such beams. By definition one has to design the machine with the SR, which increases proportionally to E4 and 1/R, where E is the energy and R the radius.  A larger machine will have less SR and require less RF voltage and power.